Finite Group Databasebeta

← C23×#13220 #19160846

Subgroups — C23×#13220 #19160846

Order 5888.

540 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2486C211yesmaximal, characteristic
2C2×C23×#352C214yesmaximal
3C23×#1800C211yesmaximal, characteristic
4C4×C23×#99C211yesmaximal, characteristic
5C2×C23×#123C2212yes—
6C2×C4×D4×C23C2211yescharacteristic
7C22×C4⋊C4×C23C411yescharacteristic, complemented
8C23×#423—21no—
9C2×C23×#100C2212yes—
10C2×C23×#99C2212yes—
11C23×#398—21no—
12C23×#397—21no—
13C23×#389—21no—
14C23×#381C412yescomplemented
15C23×#380—21no—
16C23×#380C411yescharacteristic, complemented
17C4×C22⋊C4×C23—21no—
18C23×#352—28no—
19C23×#343—21no—
20C23×#342—21no—
21C22×D4×C23C2311yescharacteristic
22C23×C4×C23C2×C412yes—
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23D411yescharacteristic
25C2×M16×C23—22no—
26C23×#123—24no—
27C23×#123—24no—
28C4×D4×C23—41no—
29C4×D4×C23—41no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—22no—
34C2×C4⋊C4×C23—41no—
35C2×C4⋊C4×C23—41no—
36C2×C4⋊C4×C23D411yescharacteristic
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23C2×C412yes—
40C2×C22⋊C4×C23C2×C411yescharacteristic
41C2×C22⋊C4×C23D411yescharacteristic
42C2×C42×C23—41no—
43C2×C42×C23D411yescharacteristic
44C23×C4⋊C8—41no—
45C23×C4⋊C8—41no—
46C23×#100—24no—
47C23×#99—22no—
48C23×#99D414yes—
49C23×#95—22no—
50C23×#95—21no—
51C23×#95—22no—
52C23×#95—41no—
53C24×C23C22×C411yescharacteristic
54C24×C23C2×D411yescharacteristic
55C2×D4×C23—41no—
56C2×D4×C23—41no—
57C2×D4×C23C2×D412yes—
58C2×D4×C23#4112yes—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—21no—
62C22×C4×C23—21no—
63C22×C4×C23—22no—
64C22×C4×C23—22no—
65C22×C4×C23—24no—
66C22×C4×C23—42no—
67C22×C4×C23—42no—
68C22×C4×C23—42no—
69C22×C4×C23—42no—
70C22×C4×C23—42no—
71C22×C4×C23C22⋊C411yescharacteristic
72C22×C4×C23C22⋊C411yescharacteristic
73C22×C4×C23C22⋊C411yescharacteristic
74C22×C4×C23C22⋊C411yescharacteristic
75C22×C4×C23C2×D411yescharacteristic
76M16×C23—42no—
77M16×C23—42no—
78C2×C8×C23—42no—
79C4⋊C4×C23—41no—
80C4⋊C4×C23—42no—
81C4⋊C4×C23—42no—
82C4⋊C4×C23—41no—
83C4⋊C4×C23—41no—
84C4⋊C4×C23—42no—
85C4⋊C4×C23—42no—
86C22⋊C4×C23—24no—
87C22⋊C4×C23—24no—
88C22⋊C4×C23—41no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—44no—
92C42×C23—21no—
93C42×C23—21no—
94C42×C23—82no—
95#13220C2311yesmaximal, characteristic, direct product factor, Sylow
96C23×C23—21no—
97C23×C23—21no—
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—41no—
101C23×C23—41no—
102C23×C23—41no—
103C23×C23—41no—
104C23×C23C2×C22⋊C411yescharacteristic
105C23×C23C4×D412yes—
106C23×C23#12112yes—
107D4×C23—42no—
108D4×C23—42no—
109C2×C4×C23—21no—
110C2×C4×C23—21no—
111C2×C4×C23—22no—
112C2×C4×C23—21no—
113C2×C4×C23—21no—
114C2×C4×C23—21no—
115C2×C4×C23—42no—
116C2×C4×C23—42no—
117C2×C4×C23—42no—
118C2×C4×C23—41no—
119C2×C4×C23—41no—
120C2×C4×C23—41no—
121C2×C4×C23—44no—
122C2×C4×C23—42no—
123C2×C4×C23—42no—
124C2×C4×C23—42no—
125C2×C4×C23—42no—
126C2×C4×C23—42no—
127C2×C4×C23—42no—
128C2×C4×C23—81no—
129C2×C4×C23—82no—
130C2×C4×C23—82no—
131C2×C4×C23—84no—
132C2×C4×C23#9911yescharacteristic
133C2×C4×C23#9911yescharacteristic
134C2×C4×C23#12411yescharacteristic
135C2×C4×C23#12711yescharacteristic
136C8×C23—82no—
137#2486C2×C2311yescharacteristic
138C2×#352C2×C2314yes—
139#1800C2×C2311yescharacteristic
140C4×#99C2×C2311yescharacteristic
141C22×C23—21no—
142C22×C23—21no—
143C22×C23—22no—
144C22×C23—22no—
145C22×C23—42no—
146C22×C23—42no—
147C22×C23—42no—
148C22×C23—41no—
149C22×C23—41no—
150C22×C23—41no—
151C22×C23—41no—
152C22×C23—81no—
153C22×C23#39011yescharacteristic
154C22×C23C2×#9911yescharacteristic
155C22×C23#41011yescharacteristic
156C4×C23—21no—
157C4×C23—21no—
158C4×C23—21no—
159C4×C23—41no—
160C4×C23—41no—
161C4×C23—41no—
162C4×C23—82no—
163C4×C23—84no—
164C4×C23—82no—
165C4×C23#35212yescomplemented
166C2×#123C22×C2312yes—
167C2×C4×D4C22×C2311yescharacteristic
168C22×C4⋊C4C4×C2311yescharacteristic, complemented
169#423—21no—
170C2×#100C22×C2312yes—
171C2×#99C22×C2312yes—
172#398—21no—
173#397—21no—
174#389—21no—
175#381C4×C2312yescomplemented
176#380—21no—
177#380C4×C2311yescharacteristic, complemented
178C4×C22⋊C4—21no—
179#352—28no—
180#343—21no—
181#342—21no—
182C2×C23—21no—
183C2×C23—21no—
184C2×C23—42no—
185C2×C23—42no—
186C2×C23#196411yescharacteristic
187C2×C23C2×#35211yescharacteristic
188C2×C23#212311yescharacteristic
189C22×D4C23×C2311yescharacteristic
190C23×C4C2×C4×C2312yes—
191C23×C4C2×C4×C2311yescharacteristic
192C23×C4D4×C2311yescharacteristic
193C2×M16—22no—
194#123—24no—
195#123—24no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4D4×C2311yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4C2×C4×C2312yes—
208C2×C22⋊C4C2×C4×C2311yescharacteristic
209C2×C22⋊C4D4×C2311yescharacteristic
210C2×C42—41no—
211C2×C42D4×C2311yescharacteristic
212C4⋊C8—41no—
213C4⋊C8—41no—
214#100—24no—
215#99—22no—
216#99D4×C2314yes—
217#95—22no—
218#95—21no—
219#95—22no—
220#95—41no—
221C23#1322011yescharacteristic, direct product factor, Sylow
222C24C22×C4×C2311yescharacteristic
223C24C2×D4×C2311yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C2×D4×C2312yes—
227C2×D4#41×C2312yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4C22⋊C4×C2311yescharacteristic
241C22×C4C22⋊C4×C2311yescharacteristic
242C22×C4C22⋊C4×C2311yescharacteristic
243C22×C4C22⋊C4×C2311yescharacteristic
244C22×C4C2×D4×C2311yescharacteristic
245M16—42no—
246M16—42no—
247C2×C8—42no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C42—21no—
262C42—21no—
263C42—82no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23C2×C22⋊C4×C2311yescharacteristic
273C23C4×D4×C2312yes—
274C23C23×#12112yes—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4C23×#9911yescharacteristic
301C2×C4C23×#9911yescharacteristic
302C2×C4C23×#12411yescharacteristic
303C2×C4C23×#12711yescharacteristic
304C8—82no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22C23×#39011yescharacteristic
318C22C2×C23×#9911yescharacteristic
319C22C23×#41011yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—84no—
328C4—82no—
329C4C23×#35212yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2C23×#196411yescharacteristic
335C2C2×C23×#35211yescharacteristic
336C2C23×#212311yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×D4×C23C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. C22×C4⋊C4×C23C4semidirect
14. C23×#381C4semidirect
16. C23×#380C4semidirect
95. #13220C23direct (N × H)
165. C4×C23#352semidirect
168. C22×C4⋊C4C4×C23semidirect
175. #381C4×C23semidirect
177. #380C4×C23semidirect
221. C23#13220direct (N × H)
329. C4C23×#352semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#24862
C2×C23×#3522
C23×#18002
C4×C23×#992
#1322023